SEMINAR SERIES ON ADVANCED MEDICAL IMAGE PROCESSING LEVEL SET METHODS: A NEW METHODOLOGY OF SEGMENTA

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SEMINAR SERIES ONADVANCED MEDICAL IMAGE
PROCESSING LEVEL SET METHODSA NEW
METHODOLOGY OF SEGMENTATION

• Mingyue Ding, PH. D.
• Robarts Research Institute
• London, Ontario, Canada
• September 27, 2002

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WHAT IS LEVEL SET?

• Level set is a set of points with same height
  such as water level or geodesic line
• Level set method as a front propagation theory
  was first proposed by Sethian in 1982
• In 1995, Malladi introduced it to image analysis
  domain, to find image boundary

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WHAT IS SEGMENTATION?

• Separate object from background
• Broadly speaking, it is to use a model whose
  boundary representation is matched to the image
  to recover the object of interest.
• Or simply, it is object recover from raw data

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HOW SNAKE WORKS?

• Initialize a guess contour clicking points in
  image
• Digitize the contour
• Move the contour under the internal and external
  forces
• Problems in snake
• Sensitive to initial guess of shape
• Difficult to recover complex structure
• Difficult to track multi-object automatically

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FRONT PROPAGATION ANOTHER UNDERSTANDING OF SNAKE

• A closed interface moving in a plane
• Or more broadly, a front moves from initial
  contour to image boundary along its normal vector
  with a speed of F
• Two different representations in front
• Parametric representation
• Level set (or geodesic ) representation

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boundary
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FRONT REPRESENTATION

• Drawbacks in 2D parametric function
• The function definition dependent on the
  different objects
• t is not a single value function when the front
  moved back and forth
• Difficult to express the complex curve
• Level set use one dimension higher function to
  represent the curve.

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LEVEL-SET SURFACE, f(x,y,z)0
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FRONT ZERO LEVEL SET

• To avoid complex 3D contour, we always suppose
  current contour has zero height. This is called
  zero level set.
• Dynamic coordinate system
• The plane of Oxy is defined dynamically
  overlapped with the evolving front.

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DETERMINATION OF IMAGE BOUNDARY

• Snake
• Determine a functional C so that
• is minimal
• Level set method
• Solve a Partial Differential Equation (PDE) , in
  which the interface is a zero level set and
  constrained by the initial contour.

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HAMILTONG-JACOBI EQUATION

• Propagating hyper-surface
• By using the chain rule, we have
• 
  (1)
• Because
• 
  (2)
• Hamilton-Jacobi equation

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SWALLOWTAIL REMOVAL

• In front propagation, a swallowtail problem in
  corner may appear when we let the boundary pass
  itself
• Huygens principle construction or a entropy
  satisfying solution, i.e., we only expand the
  boundary which consists of the points located a
  distance, t ,from the initial curve

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NUMERICAL APPROXIMATION

• Suppose we use a uniform mesh of spacing h and a
  time step of , the Hamilton-Jacobi equation
  will be
• where is the appropriate finite
  difference operator for the spatial derivative

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HOW TO DETERMINE THE SPEED?

• The normal vector speed FF(L,G,I) is determined
  by
• Local properties such as curvature and normal
  direction
• Global properties of the front like PDE
• Independent properties. For instance an
  underlying fluid velocity. R(x,y) 2 I(x,y)-1.

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DETERMINATION OF SPEED

• We invoke the ENTROPY CONDITION and HYPERBOLIC
  CONSERVATION LAWS
• where K is the curvature of hyper-surface,
• F0 is a constant inflation term and F1 (K) is a
  term depending on the geometry of front

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DEFINITION OF SPEED TERMS

• For example, we can choice
• where is a constant acted as an
  advection term while the uniform expansion speed,
  1 (or -1), corresponds to the inflation (or
  shrink) force.

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DEMO OF LEVEL SET MOVING
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FRONT STOPPING CRITERION

• In order to let the front halting on the
  boundary, we must define such a speed that acts
  as a stopping criterion for this speed function
  by multiplying the term

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DEFINITION OF STOPPING CRITERIONS

• Different definitions of stopping term

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ORIGINAL IMAGE
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(a) sigma0.3
EFFECT OF THE VALUES OF SIGMA
(b) sigma0.5
(c) sigma1.0
(d) sigma2.0
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(a) Reciprocal function(N1)
IMAGE-BASED SPEED COMPARISON
(b) Reciprocal function(N2)
(c) Exponential function
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EXTENDING SPEED FUNCTION

• The speed is locally defined along the boundary
  but not globally defined
• Requirements for extension
• Level set moving under this speed function cannot
  collide
• Computation efficient

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SPEED EXTENSION PROBLEM
Speed-defined points
Initial contour
Speed-undefined points
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EXTENDING SPEED FUNCTION

• There are different ways to extend the speed
  function to the neighboring level sets
• Global extension nearest speed point
• Global extension with re-initialization
• Narrow-band extension
• Narrow-band extension with re-initialization

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NUMERICAL SOLUTION OF HAMILTON-JACOBI EQUATION

• We can get the entropy-satisfying weak solution
  of Hamilton-Jacobi equation by the following
  iteration

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NUMERICAL SOLUTION OF HAMILTON-JACOBI EQUATION

• Similarly, in 2-D case, the solution is

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FINDING THE FRONT, X(t)

• Given a cell of (i,j), if
• the cell cannot contain the front X(t)
• Otherwise, find the entrance and exit points by
  linear interpolation which is one of our
  approximation to X(t)
• Collection of all such line segments consists of
  our approximation to X(t)

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INNER (HOLE) BOUNDARY SEGMENTATION

• Temporarily relax the stop criterion and allow
  the front to move past the outer boundary
• Once it occurs, the stopping criterion is turned
  back on.
• Resume the level set front evolving

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FAST MARCHING METHODS

• In level set methods, in order to avoid the
  missing of boundary, a very small time step
  should be adopted, leading a large number of
  iterations.
• Fast marching methods can be used to greatly
  accelerate the initial propagation from the seed
  structure to the near boundary

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LEVEL SET SEGMENTATION ALGORITHM

• 1 Initialize a contour X0
• 2 Calculate the speed along X0
• 3 Extend the speed calculation
• 4 Level set function calculation
• 5 Find the evolving front
• 6 If speed is near 0, stop. Otherwise go to Step
  2
• 7 If no front point moved, end the segmentation

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ARTERY BOUNDARY TRACKING
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CONTOUR DETECTION BY CLICKING
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NOISE REMOVAL WITH EDGE PRESEVING
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ITK

• LevelSetCurvatureFunction (itk)   
• LevelSetFunction (itk)   
• LevelSetFunctionGlobalDataStruct (itk)   
• LevelSetFunctionBase (itk)   
• LevelSetImageFilter (itk)   
• LevelSetNeighborhoodExtractor (itk)   
• LevelSetNode (itk)   
• LevelSetTypeDefault (itk)     
• LevelSetVelocityNeighborhoodExtractor (itk) 

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CONCLUSION

• Level set is a new methodology for segmentation
  and different application. It has the following
  features
• Insensitive to the initial contour guess
• Fast and easy to be extended to high dimension
• Complex topological structure
• Can be processed in parallel

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CONCLUSION

• Open problems
• Sensitive to sharp corners, cusps and topological
  changes
• Segmentation result greatly depending on the
  design of stopping criteria
• Complexity in speed extension

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BOOKS

• J. A. Sethian, An Analysis of Flame Propagation,
  Ph. D. Dissertation, Dept. of Math, University of
  California, Berkeley, CA, 1982
• J. A. Sethian, Level Set Methods, Cambridge
  University Press, 1996
• J. A. Sethian, Level Set Methods and Fast
  Marching Methods, Cambridge University Press,
  1999, 2000, 2001
• S.J. Osher, R.P. Fedkiw, Level Set Methods and
  Dynamic Implicit Surfaces, Springer Verlag, 2002

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WEBSITES

• //math.berkeley.edu/sethian/level_set.html
• http//www.math.ucla.edu/sjo/
• http//www.levelset.com/lss.html

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PAPERS

• J.A.Sethian, et al., Crystal growing and
  dendritic solidification, Journal of
  Computational Physics, Vol. 98, 231-253,1992
• R. Malladi, et al., Shap Modeling with Front
  Propagation A Level set Approach, IEEE Trans.
  PAMI-17, 158-175,1995
• -, A Unified Approach to Noise Removal, Image
  Enhancement, and Shape Recovery, IEEE Trans.
  IP-5,1554-11568,1996
• V. Caselles, et al., Geodesic Active Contours,
  Inter. J of Computer vision, Vol. 22(1), 61-79,
  1997
• Tony F. Chan, and L.A. Vese, Active Contours
  Without Edges, IEEE Trans. IP-10,266-277,2001
• R. Goldberg, et al, Fast Geodesic Active
  Contours, IEEE Trans. IP-10, 1467-1475, 2001
• E. Debreuve, et al., Space-Time Segmentation
  Using Level Set Active Contours Applied to
  Myocardial Gated SPECT, IEEE Trans. MI-20,
  643-659, 2001